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A360857 Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).

%I A360857 #11 Mar 06 2023 09:51:59
%S A360857 1,1,1,1,2,6,1,3,12,12,1,4,20,30,60,1,5,30,60,150,150,1,6,42,105,315,
%T A360857 420,700,1,7,56,168,588,980,1960,1960,1,8,72,252,1008,2016,4704,5880,
%U A360857 8820,1,9,90,360,1620,3780,10080,15120,26460,26460
%N A360857 Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).
%e A360857 Table T(n, k) starts:
%e A360857 [0] 1;
%e A360857 [1] 1, 1;
%e A360857 [2] 1, 2,  6;
%e A360857 [3] 1, 3, 12,  12;
%e A360857 [4] 1, 4, 20,  30,   60;
%e A360857 [5] 1, 5, 30,  60,  150,  150;
%e A360857 [6] 1, 6, 42, 105,  315,  420,   700;
%e A360857 [7] 1, 7, 56, 168,  588,  980,  1960,  1960;
%e A360857 [8] 1, 8, 72, 252, 1008, 2016,  4704,  5880,  8820;
%e A360857 [9] 1, 9, 90, 360, 1620, 3780, 10080, 15120, 26460, 26460.
%p A360857 A360857 := (n, k) -> binomial(n, ceil(k/2))*binomial(n + 1, floor(k/2)):
%p A360857 seq(seq(A360857(n, k), k=0..n), n=0..9);
%t A360857 Table[Binomial[n,Ceiling[k/2]]Binomial[n+1,Floor[k/2]],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Mar 06 2023 *)
%o A360857 (Python)
%o A360857 from math import comb
%o A360857 def A360857_T(n,k): return comb(n+1,m:=k>>1)**2*(n+1-m)*(n-m)//((m+1)*(n+1)) if k&1 else comb(n+1,m:=k>>1)**2*(n+1-m)//(n+1) # _Chai Wah Wu_, Feb 28 2023
%Y A360857 Cf. A360858, A360859.
%K A360857 nonn,tabl
%O A360857 0,5
%A A360857 _Peter Luschny_, Feb 28 2023